Prepare the pasta dough according to the recipe and roll out to the thinnest setting on a pasta-rolling machine. Using a paring knife, cut the pasta sheets into long strands of spaghetti. Carefully hang the strands over a dowel over the back of a chair to dry.

Bring 6 quarts of water to a boil and add 2 tablespoons salt.

Meanwhile, in a large, heavy-bottomed skillet, heat the olive oil over medium-high heat and add the garlic. Cook for 2 minutes, until it is fragrant and golden, then add the snails and mint, and toss well. Season with salt and pepper and then add the oregano. Reduce the heat to medium-low and cook for 10 to 12 minutes.

Once the snails are cooked through and fragrant, cook the spaghetti in the boiling water until tender yet al dente, about 4 minutes. Drain and add to the pan with the snails. Gently toss over high heat for 1 minute, adding more olive oil or a little pasta cooking water if necessary, and divide evenly among 6 warmed pasta bowls.